Download 5. The Hydrogenoid Atom

CHEM 333
Quantum Theory and Spectroscopy
5. THE HYDROGENOID ATOM: ELECTRONIC STRUCTURE AND SPECTRA
5.1. ENERGY AND WAVEFUNCTION
♦ The SCHROEDINGER equation for the Hydrogenic atom (Z protons, 1 electron)
V (r ) =
− Ze 2 − Ze' 2
=
4πε 0 r
r
− 2 2 ˆ
Hˆ =
∇ +V
2µ
Hˆ ψ = Eψ
− 2 2 ˆ − 2 1 ∂  2 ∂ 
Lˆ2  − Ze' 2
ˆ
+
∇ +V = 
H=
r
+
2
2 
2µ
r
 2 µ r ∂r  ∂r  2 µr 
The form of the Hamiltonian suggests a separation of variables:
where
in spherical coordinates.
ψ (r ,θ , φ ) = R(r )Yl m (θ , φ )
Yl m (θ , φ ) are the spherical harmonics – eigenfunctions of the L̂2 operator.
Upon substitution of
ψ (r ,θ , φ ) into the SCHROEDINGER equation, one obtains a radial equation
d 2 R 2 dR  2 E 2 Z l (l + 1) 
+
+
+
−
R=0
dr 2 r dr  ae' 2 ar
r 2 
R(r ) ~ e − Cr
♦ Asymptotic solution of the radial equation (large r )
♦ We thus assume a wavefunction of the form
a=
with
C=
2
µe ' 2
− 2E
ae' 2
(E
<0)
R(r ) = e − Cr K (r )
and we solve the radial equation for K ( r ) by the power series method. Actually,
∞
K (r ) = r l ∑ b j r j .
j =0
-25-
CHEM 333
Quantum Theory and Spectroscopy
b j +1 =
We end up with the recursion relation
♦ Boundary conditions
j,
For large
b j +1
bj
~
2C
j
(2C + 2Cl + 2Cj − 2Z / a)
bj
j ( j + 1) + 2(l + 1)( j + 1)
R(r ) = e −Cr r l e 2Cr = r l e Cr
K (r ) = r l e 2Cr
In order for the wavefunction to be asymptotically finite (and then be well-behaved), we must truncate the power
series after a finite number of terms, say for j
= k . We then obtain
n = k + l +1
We introduce a new integer as
Cn = Z / a together with C =
2C (k + l + 1) = 2 Z / a
n = 1,2,3,...
− 2E
yields:
ae' 2
E=−
Z2
n2
l ≤ n −1
 e' 2 
Z 2 µe 4 1
  = −
2
2
8ε 0 h 2 n
 2a 
We have derived the same quantized energy expression as that of the BOHR classical model of the Hydrogenic
atom, but this time it was derived rigorously from Quantum Mechanics. The quantization once again results from
imposing boundary conditions on the wavefunction.
♦ The general solutions of the radial equation for the Hydrogenic atom are of the form
Rnl (r ) = r l e −Cr
n − l −1
∑b r
j
j
= N nl r l e − Zr / na Lnl (r )
(C
= Z / na )
j =0
where
N nl is a normalization constant and Lnl are the (well-tabulated) associated LAGUERRE polynomials.
♦ The wavefunctions of the Hydrogenic atom depend on 3 quantum numbers and have the general form
ψ nlm (r ,θ , φ ) = Rnl (r )Yl m (θ , φ )
n = 1,2,3,...
-26-
l ≤ n −1
m ≤l
CHEM 333
Quantum Theory and Spectroscopy
§ The constant a in the wavefunction expression for the Hydrogenic atom is closely related to the
BOHR radius. Replacing the proton-electron reduced mass
a0 =
a , one obtains
µ
by the electron mass in the expression for
4πε 0 2
2
=
= 0.52918Å
me e ' 2
me e 2
§ Degeneracy of energy levels. A given energy level of the Hydrogenic atom (i.e. a given n ) has a
n −1
degeneracy of
l
n −1
∑ ∑1 =∑ (2l + 1) = n 2
l =1 m = − l
l =1
5.2. ELECTRONIC STRUCTURE AND ORBITALS
♦ One-electron wavefunctions for electrons in atoms are called orbitals. Hydrogenic atom orbitals are
characterized by 3 quantum numbers
n, l , m . The electron state in denoted n, l , m .
E = −Z 2 RH / n 2
*
n is the orbital principal quantum number and characterizes the electron energy
*
l is the orbital angular momentum quantum number and characterizes the electron orbital angular
momentum, which is of magnitude
*
l (l + 1)
l = 0,1,2,3,..., n − 1
with
m is the orbital magnetic quantum number and characterizes the z-component of the electron orbital
angular momentum, which can take on values
m
with
m = 0,±1,±2,±3,...,±l
♦ Electrons also possess an intrinsic angular momentum characterized by two quantum numbers s and ms (the
analogs of
l and m ). s is fixed at 1 / 2 and ms can take on values − 1 / 2 or 1 / 2 .
-27-
CHEM 333
Quantum Theory and Spectroscopy
♦ All orbitals of a given value n form a single shell of the atom. Each shell is n 2 -degenerate.
Orbitals with a given value
with the letters
n and different values of l form a subshell of a given value. Subshells are referred to
s (l = 0) , p (l = 1) , d (l = 2) , f (l = 3) ,etc.
Orbital diagram (each box corresponds to an orbital with a different magnetic quantum number):
s
p
n=3
‫‫‬
‫‫‫‫‬
n=2
‫‫‬
‫‫‫‫‬
n =1
‫‫‬
♦ The square of an orbital ψ
2
d
‫‫‫‫‫‫‬
3s3 p3d
2s 2 p
1s
represents the electron probability density. Orbitals are usually plotted as
surfaces that capture, say 90%, of the electron density.
♦ Probability of finding an electron in space for a given orbital:
P(r ,θ , φ )dτ = ψ 2 (r ,θ , φ )r 2 sin θdrdθdφ
ψ (r ,θ , φ ) = Rnl (r )Yl m (θ , φ )
Radial probability (after integrating over angular coordinates):
P (r )dr = Rnl2 (r )r 2 dr
∞
Mean radius of the electron:
r = ∫ rP(r )dr
Most probable electron radius:
0
§ Electron in the 1s orbital
r* =
a0
Z
-28-
r =
3 a0
2 Z
 dP(r ) 
=0
 dr 
r = r*
CHEM 333
Quantum Theory and Spectroscopy
Hydrogenoid Radial Wavefunctions
Orbital
1s
2s
2p
3s
3p
3d
n
1
2
2
3
3
3
l
Rnl
0
Z
2 
 a0 
0
1 Z
 
2 2  a 0 
1
1 Z

4 6  a 0



0
1 Z

9 3  a0



1
Z

27 6  a 0
2
Z

81 30  a 0
1
1
3/ 2
e −ρ / 2
3/ 2
(2 −
1
ρ ) e −ρ / 4
2
3/ 2
ρ e−ρ / 4
3/ 2
(6 − 2 ρ +



3/ 2



1 2 −ρ / 6
ρ )e
9
1
(4 − ρ ) ρ e − ρ / 6
3
3/ 2
ρ 2 e −ρ / 6
ρ = 2Zr / a0
Taken from Physical Chemistry, Alberty and Silbey, Wiley, 1996.
-29-
CHEM 333
Quantum Theory and Spectroscopy
5.3. QUANTUM NUMBERS AND SPECTROSCOPIC SELECTION RULES
♦ A transition from one electron orbital quantum state to another results from absorption or emission of a photon
hν
i = ni , li , mi ±→

n f ,l f , m f = f
(light)
♦ Only spectroscopic transitions with nonzero transition dipole moment µ tr = i µˆ f are allowed.
→
Selection rules for atomic transitions:
∆l = ±1
∆m = 0,±1
The selection rules are consistent with total angular momentum conservation and the fact that a
photon has an intrinsic angular momentum
lp =1
♦ Zeeman effect: a magnetic field splits the orbital energy levels of a (sub)shell
In the presence of a magnetic field
B z , the total Hamiltonian becomes
The energy of an electron in orbital state
n, l , m becomes
eB
Hˆ = Hˆ 0 + z Lˆ z
2m e
E = − Z 2 RH
1
+ βmB z
n2
where
β
is the
Bohr magneton ( β
= 9.274 × 10 −24 J ⋅ T −1 ). Because a magnetic field splits the 2l + 1 levels of a (sub)shell
characterized by the
m quantum number, m is called the magnetic quantum number.
§ Nuclear Magnetic Resonance and Electron Spin Resonance spectroscopies also arise from the fact
that the energy levels of nuclear spin and electron spin are perturbed by a magnetic field, respectively.
-30-